Kolmogorov Equation for a Stochastic Reaction–Diffusion Equation with Multiplicative Noise

Kaiyuqi Guan and Yu Shi ()
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Kaiyuqi Guan: School of Mathematics and Statistics, Wuhan University of Technology, Wuhan 430062, China
Yu Shi: School of Mathematics and Statistics, Wuhan University of Technology, Wuhan 430062, China

Mathematics, 2025, vol. 13, issue 10, 1-19

Abstract: Reaction–diffusion equations can model complex systems where randomness plays a role, capturing the interaction between diffusion processes and random fluctuations. The Kolmogorov equations associated with these systems play an important role in understanding the long-term behavior, stability, and control of such complex systems. In this paper, we investigate the existence of a classical solution for the Kolmogorov equation associated with a stochastic reaction–diffusion equation driven by nonlinear multiplicative trace-class noise. We also establish the existence of an invariant measure ν for the corresponding transition semigroup P t , where the infinitesimal generator in L 2 ( H , ν ) is identified as the closure of the Kolmogorov operator K 0 .

Keywords: kolmogorov equation; stochastic reaction–diffusion equation; multiplicative noise; invariant measure; transition semigroup (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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